The claim that maybe stood up was, as you say, not clear which variable was involved. That's not the same as saying "the original non-ionizing claim associated with power lines about childhood leukaemia stood up".
The original claim was by an epidemiologist, I forget her name, that there was a corrolation between leukemia and power lines. Now, the original study was done with either no funding, or very little, by a lone researcher guestimating the distance between power lines and the homes one kids with leukemia. All the usual things about clusters of cancer cases and texas sharp shooters apply. Somehow the researcher lucked out though and 40 odd years on there is still something borderline worth talking about.
The original claim then got blown out of all porportion by journalists, fringe scientists and people with "alternative" beliefs.